UPSC 2024 Maths Optional Paper 1 Q8b — Step-by-Step Solution

15 marks · Section B

Gauss divergence theorem · Vector Analysis · asked 10× in 14 yrs · Read the full method →

Question

Using Gauss’s divergence theorem, evaluate

\iint_S (y^2\hat i+xz^3\hat j+(z-1)^2\hat k)\cdot\hat n\,dS

over the region bounded by the cylinder $x^2+y^2=16$ and the planes $z=1$ and $z=5$ .

Technique

Compute $\nabla\cdot\vec F$ (only the third component contributes); switch to cylindrical coordinates; the integral separates by Fubini.

Solution

$The closed region for the divergence theorem: the solid cylinder x^2+y^2\le16 (radius r=4) between the planes z=1 and z=5, with outward unit normal \hat n on the curved side, the top disk, and the bottom disk. Converting the surface flux to the volume integral of \nabla\cdot\vec F over this region, in cylindrical coordinates, is the whole method.$

Step 1 — Divergence.

\vec F=(y^2,\,xz^3,\,(z-1)^2),\qquad\nabla\cdot\vec F=0+0+2(z-1)=2(z-1).

Step 2 — Volume integral (divergence theorem).

\iint_S\vec F\cdot\hat n\,dS=\iiint_V 2(z-1)\,dV.

Volume $V$ : cylinder $r\le 4$ , $1\le z\le 5$ . In cylindrical coordinates $dV=r\,dr\,d\theta\,dz$ .

Step 3 — Evaluate.

The integral separates:

\iiint_V 2(z-1)\,dV=\underbrace{\int_0^{2\pi}d\theta}_{2\pi}\cdot\underbrace{\int_0^4 r\,dr}_{8}\cdot\underbrace{\int_1^5 2(z-1)\,dz}_{16}.

$\int_1^5 2(z-1)\,dz=[(z-1)^2]_1^5=16$ .

\text{Total}=2\pi\cdot 8\cdot 16=256\pi.

Answer

\boxed{\;256\pi.\;}